President Barack Obama and congressional Republican leaders are finally prepared to open negotiations this week on the fiscal cliff — an all-important legislative riddle with no easy answers.

Democrats want to increase tax rates; Republicans say no way. Republicans seek major changes to entitlement programs; Democrats are divided. Republicans want to nullify a package of automatic spending cuts; Democrats won’t go for it without major concessions.

The policy differences are endless, but the scenarios for how it could play out are fairly limited. Here are five ways Congress may navigate the labyrinth:

1. Go over the cliff

In gridlocked Washington, this outcome is not only possible, it’s the preference of some in both parties.

Consider the reasons, from the practical to the political.

There just isn’t a lot of time before Jan. 1. Congress returns this week but then leaves town for Thanksgiving. After returning from that break, lawmakers will have only four weeks to strike a compromise on issues that have confounded the parties for years.

Congress usually delivers only when it’s up against a deadline, but if lawmakers do so now they could be viewed as jamming a complex bill into law without much scrutiny.

House Speaker John Boehner (R-Ohio) has said he doesn’t want retiring and defeated lawmakers to vote on major legislation during the lame-duck session. He would rather temporarily extend all of the tax rates to allow more time next year to overhaul the Tax Code. But Democrats don’t want to renew the rates for the top earners, even for a few months.

Even if the two sides can come to an agreement on how to handle income tax rates, they’ll have to develop a framework for comprehensive tax reform — and there’s almost no agreement on what that should look like.

While this debate is raging, Congress also has to deal with other items that aren’t getting a lot of attention but could cause big problems. The pricey alternative minimum tax, which will hit millions of upper-middle-class families, needs to be patched. House Republican leaders want to extend the nation’s farm policy. Corporate tax breaks are about to expire. There’s also the debt ceiling, which will need to be raised in early 2013. And don’t forget the automatic spending cuts to the Pentagon and domestic programs — known as the sequester — which both parties want to avoid but need to find a way to pay for.

Top Democrats see a strategic advantage in letting the rates lapse. The logic is that, once taxes rise Jan. 1, lawmakers would then be able to say they voted to cut taxes. This could make it easier to attract Republicans who may otherwise feel locked into Grover Norquist’s no-tax-hike pledge.

Another incentive for Senate Democrats is that they are expected to pick up two seats in the next Congress, giving them a slightly larger majority to negotiate a more favorable deal.

The economy could drop into a recession by the end of next year if nothing is done. The impact would be significant in the short term — because taxes would rise on every single American — but not catastrophic, some analysts say, as long as Congress reaches agreement within a few weeks and reverses the damage.

The wild card is the president and a question only he can answer if the talks falter: Is it worse to go over the cliff or renew tax cuts for the wealthy?

“We just had an election where President Obama ran on that. We increased our majority in the Senate with Democratic candidates who said that to solve this problem the wealthiest Americans have to pay their fair share, too,” said Sen. Patty Murray (D-Wash.), a member of Senate leadership, on ABC’s “This Week.” “If the Republicans will not agree with that, we will reach a point at the end of this year where all the tax cuts expire, and we will start over next year and whatever we do will be a tax cut for whatever package we put together. That may be a way past this.”

2. A big deal

This is what Boehner wants: a framework that would force Congress to spend 2013 reforming the Tax Code and stabilizing entitlement programs.

Boehner said last week that “2013 should be the year we begin to solve our debt through tax reform and entitlement reform. And I’m proposing that we avert the fiscal cliff together in a manner that ensures that 2013 is finally the year that our government comes to grips with the major problems that are facing us. This will bring jobs home, result in a stronger and healthier economy, and a stronger and healthier economy means more Americans working and more revenues, which is what the president is seeking.”

A grand bargain is also Obama’s first choice. It would allow him to sweep away these major issues at the start of his second term, get the economy moving again and focus on other issues.

But it won’t be easy.

Developing a framework for a big deal would mean the two parties would have to agree on how much revenue tax reform should generate, the rate structure, which entitlements would be tweaked and a deadline for doing so. The two sides are in opposite corners on nearly every aspect of tax reform. Republicans want to lower rates; Democrats want to increase them. Republicans say lower rates would result in more revenue for the government; Democrats call that voodoo math. They don’t even agree on how budget scorekeepers should judge the impact of tax reform.

They could agree to raise the income threshold for the tax increases to $500,000 or $1 million and pick up the rest of the revenue through loophole closures. But a deal like this, which was promoted Sunday by a leading conservative, Week Standard editor Bill Kristol, would mean both sides would have to abandon their hardened positions.

There are a multitude of peripheral issues. There would need to be an enforcement mechanism that kicks in if Congress fails to act. And any framework would need to include expedited instructions, so legislation wouldn’t get bogged down by procedural games in the Senate.

It would also become a bonanza for K Street and corporate America, which would use the next seven weeks to protect their interests.

Then there are entitlements. Both parties need to agree how far to cut treasured programs such as Medicare and Medicaid. Senate Majority Leader Harry Reid has said he won’t “mess with Social Security” even though Republicans and Democrats have considered changes to the program.

Both parties think they have a good outline: Obama and Boehner’s framework from their 2011 talks. But as the speaker frequently says, it remains to be seen whether Humpty Dumpty can be put back together again.

3. Tax-free strategy

Given that tax cuts are the major sticking point, why not just remove them from the end-of-year equation?

Democrats could agree to the Republican demands to lift the sequester for a year if it is paid for in a balanced way. In return, Democrats would seek some kind of tax break along the lines of the payroll tax cut, which expires Dec. 31. The package could also deal with other unfinished business, like unemployment benefits, the automatic rate decrease for Medicare providers and the Alternative Minimum Tax.

If Congress were to address this list of items before Jan. 1, the cliff wouldn’t be so steep, mitigating the economic shock.

Even though Obama has called on Congress to deal with the middle-class tax cuts now, Republicans see no incentive so far to separate them from the rest of the debate.

4. Buy more time

This is the classic Washington solution: When a permanent fix proves too difficult, patch it up for a few months.

Congress has been doing this for years with the federal budget. So don’t be surprised if a growing number of lawmakers call for a six-month or yearlong extension of the status quo to give them more time to deal with it.

“The watchword up there might be ‘do no harm,’” said Jim Dyer, a former Republican House Appropriations Committee staff director and top congressional liaison for Presidents Ronald Reagan and George H.W. Bush. “Let’s at least not let things happen that damage the economy.”

At this point, Obama isn’t buying it.

He already agreed to one temporary extension, in 2010, of all the Bush-era tax rates. And he’s under immense pressure not to do it again.

The White House points to a report last week from the nonpartisan Congressional Budget Office. It concluded that renewing the high-end tax cuts would have a smaller impact on job growth than Republicans claim. The West Wing sees that report as supporting a central argument: The country is spending a whole lot of money on tax cuts and can get a better bang for its buck by funneling it into an investment that creates more jobs.

“You should not accept a bad deal of extending everything for a while,” said Chad Stone, chief economist at the liberal Center on Budget and Policy Priorities. “It is bad budget policy to extend them under any circumstances.”

This outcome also could be panned by the credit-rating agencies. They have already warned that simply postponing the tough decisions could lead to the second downgrade in two years — an almost unthinkable scenario before the debt ceiling showdown of 2011.

5. Buy more time — but with a down payment

Call it the multistage strategy.

Congress buys itself a few more months or a year but couples it with a small package of spending cuts and revenue increases to signal to Wall Street that it isn’t simply avoiding tough choices.

But the challenge is the same as — or maybe even greater than — the other options.

Democrats won’t agree to spending cuts alone, so they will need to persuade Republicans to kick in some revenue, most likely by closing loopholes. Boehner is now open to that, which removes a huge obstacle, but it’s often more difficult for Congress to reach small deals. That’s because the big deals spread the pain more evenly.

A short-term deal, even with a down payment, would likely mean Obama reneges on ending the high-end tax cuts once and for all Jan. 1.

Which would put the White House back where it started, deciding the political cost of the president’s campaign pledge.